Created for C.V.C.A. Physics by Dick Heckathorn 4 October 2K+4 Vector Addition Created for C.V.C.A. Physics by Dick Heckathorn 4 October 2K+4

Vector Addition In this problem we will add three displacement vectors, first by using a scale drawing and then by using a TI-83+ calculator program called ‘VADD’.

The Problem A police cruiser chasing a speeding motorist traveled 60 km [S], then 35 km [E 45o N], and finally 50 km [W]. (a) What is the total displacement of the cruiser? Practice problem #1, page 94 from Fundamentals of Physics: Combined Course

Selecting a Scale Select a scale that will allow you to draw an arrow to represent each displacement. This scale must be large enough to be easy to work with, yet not so large that the drawing will exceed the work area.

Selecting a Scale Possible scale: 1 cm = 10 km A metric scale will be most useful because of its divisibility by ten. Possible scale: 1 cm = 10 km

Drawing the Vectors and finding the resultant

Represent the third displacement 50 km [W] Drawing Vectors 1 cm = 10 km W E Represent the third displacement 50 km [W] Represent the second displacement 35 km E 45o N Represent the first displacement 60 km [S] orientated in a north-south orientation with the arrow at the south end. Draw a line from the tail of the first vector to the head of the last vector. by an arrow 3.5 cm long, with its tail connected to the head of the first vector. Place an arrow on the end of the line where the line meets the head of the third vector. Resultant 1 N 3 by an arrow 6 cm long by an arrow 5 cm long in a west direction. E S 2 W E

Find Resultant Displacement 1 cm = 10 km N Find Resultant Displacement W E Resultant Measure the length of the line with your ruler Convert this length to the magnitude of the resultant. Measured length is 4.3 cm The resultant is 43 km

Find Resultant Direction 1 cm = 10 km N Find Resultant Direction W E Resultant Then express the direction using proper notation. Measure the angle that the resultant is from south. Measured angle is 43 km 36o S 36o W

The displacement of the cruiser is: 43 km S 36o W

Finding Average Speed The total distance traveled is: 60 km + 35 km + 50 km = 145 km The time of travel is: 1.3 hrs

Finding Average Velocity The time of travel is: 1.3 hrs The displacement is: 43 km S 36o W

Note... The vector quantity, displacement, was divided by a scalar quantity, time. When dividing a vector quantity by a scalar quantity, the answer is also a vector quantity with the same direction as the original vector quantity.

Using ‘Vector’ Program 1. ‘PROG’, ‘VADD’, ‘ENTER’, ‘ENTER’ 2. Select: 1: ADD VECTORS, ‘ENTER’

Using ‘Vector’ Program 3. NUMBER OF VECTORS: Enter ‘3’ 4. Note...one will need to know the magnitude and direction of each vector.

Determining Direction All vector directions must be specified as relative to a reference direction. Use ‘East’ as your reference direction.

Determining Direction Angles measured counter-clockwise will be identified as positive and angles measured clockwise will be identified as negative.

Determining Direction To determine the direction of each vector, first draws the coordinates.

N W E S

Determining Direction Next, sketch each vector on the coordinate reference frame with the tail of each vector at the origin of the coordinate system. Finally, determine the number of degrees each vector is from the reference direction which we have selected as East.

N 45o 2 3 180o W E 1 -90o S

Enter Vector Data Into the Calculator When prompted for magnitude and direction, enter the following: R Angle #1 60, -90o #2 35, 45o #3 50, 180o

Interpreting the Information When ‘Enter’ is pressed, one should see the three vectors drawn on the screen. Take time to inspect the drawn vectors to see that they agree with reality.

Interpreting the Information When ‘Enter’ is pressed again, the resultant vector is drawn.

Interpreting the Information When ‘Enter’ is pressed again, the magnitude and direction of the resultant is shown as: 43.4 (km), -125.6o

Interpreting the Information Convert the direction to desired notation of -125.6o : Reference direction, angle, direction S 35.6o W The displacement is then: 43.4 km, S 35.6o W

Interpreting the Information Once the displacement is found, one finds the average speed and average velocity as discussed previously. That is all that there is to it.

That’s all folks!